Image definition evaluation functions for X-ray crystallography: a new perspective on the phase problem

The core theme of X-ray crystallography is reconstructing the electron density distribution of crystals under the constraints of observed diffraction data. Nevertheless, the reconstruction of electron density distribution by straightforward Fourier synthesis is usually hindered due to the well-known phase problem and finite resolution of diffraction data. In analogy with optical imaging system, the reconstructed electron density map may be regarded as the image of the real electron density distribution in crystals. Inspired by image definition evaluation functions applied in auto-focusing process, we proposed two evaluation functions for the reconstructed electron density images. One of them is based on atomicity of electron density distribution and properties of Fourier synthesis. Tests were performed on synthetic data of known structures, and it was found that this evaluation function can distinguish the correctly reconstructed electron density image from wrong ones when diffraction data of atomic resolution is available. An algorithm was established based on this evaluation function and applied in reconstructing the electron density image from synthetic data of known structures. The other evaluation function, which is based on the positivity of electron density and constrained entropy maximization, was designed for cases where only diffraction data of rather limited resolution is available. Tests on synthetic data indicate that this evaluation function may identify the correct phase set even for a dataset at the resolution as low as 3.5 {\AA}. Though no algorithm of structure solution has been figured out based on the latter function, the results presented here provide a new perspective on the phase problem.